Math Problem Statement
Solution
The given problem is to find the value of the series:
To evaluate this, let's examine each term in the series. Notice that each term has the form:
where follows a pattern of values: .
Step 1: Simplify Each Term
We can simplify each term by using partial fraction decomposition:
Multiplying through by gives:
Expanding and solving for and :
Equating terms, we get:
From , we find . Substituting into , we get .
Thus:
Step 2: Rewrite the Series
Using this decomposition, the series becomes:
Step 3: Observe the Telescoping Nature
Notice that the terms cancel out in a telescoping manner, leaving only the first and the last terms:
Step 4: Simplify the Remaining Expression
Calculating the remaining expression:
Simplifying further:
Final Answer
The value of the series is:
Would you like a further breakdown of these steps?
Here are some related questions for further exploration:
- How would you approach a similar series with different constants in the denominator?
- Can you generalize the formula for the sum of a series in the form ?
- How would the answer change if the series extended to infinity?
- What would happen if the numerator had a different value than 7?
- Can you derive a telescoping series for other patterns of terms?
Tip: Telescoping series often simplify complex sums by allowing terms to cancel out, leaving only a few terms for calculation.
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Math Problem Analysis
Mathematical Concepts
Series Summation
Telescoping Series
Partial Fraction Decomposition
Formulas
Partial fraction decomposition: 7/(n(n+3)) = 7/3 * (1/n - 1/(n+3))
Theorems
Telescoping Series
Suitable Grade Level
Grades 11-12
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